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G = C32×Dic3order 108 = 22·33

Direct product of C32 and Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C32×Dic3, C333C4, C324C12, C3⋊(C3×C12), C6.(C3×C6), (C3×C6).7C6, C2.(S3×C32), C6.10(C3×S3), (C3×C6).11S3, (C32×C6).1C2, SmallGroup(108,32)

Series: Derived Chief Lower central Upper central

C1C3 — C32×Dic3
C1C3C6C3×C6C32×C6 — C32×Dic3
C3 — C32×Dic3
C1C3×C6

Generators and relations for C32×Dic3
 G = < a,b,c,d | a3=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 80 in 52 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3 [×4], C3 [×4], C4, C6, C6 [×4], C6 [×4], C32, C32 [×4], C32 [×4], Dic3, C12 [×4], C3×C6, C3×C6 [×4], C3×C6 [×4], C33, C3×Dic3 [×4], C3×C12, C32×C6, C32×Dic3
Quotients: C1, C2, C3 [×4], C4, S3, C6 [×4], C32, Dic3, C12 [×4], C3×S3 [×4], C3×C6, C3×Dic3 [×4], C3×C12, S3×C32, C32×Dic3

Smallest permutation representation of C32×Dic3
On 36 points
Generators in S36
(1 15 7)(2 16 8)(3 17 9)(4 18 10)(5 13 11)(6 14 12)(19 33 29)(20 34 30)(21 35 25)(22 36 26)(23 31 27)(24 32 28)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)

G:=sub<Sym(36)| (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33)>;

G:=Group( (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33) );

G=PermutationGroup([(1,15,7),(2,16,8),(3,17,9),(4,18,10),(5,13,11),(6,14,12),(19,33,29),(20,34,30),(21,35,25),(22,36,26),(23,31,27),(24,32,28)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)])

C32×Dic3 is a maximal subgroup of   C338(C2×C4)  C338D4  C334Q8  S3×C3×C12  He3⋊C12

54 conjugacy classes

class 1  2 3A···3H3I···3Q4A4B6A···6H6I···6Q12A···12P
order123···33···3446···66···612···12
size111···12···2331···12···23···3

54 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12S3Dic3C3×S3C3×Dic3
kernelC32×Dic3C32×C6C3×Dic3C33C3×C6C32C3×C6C32C6C3
# reps11828161188

Matrix representation of C32×Dic3 in GL3(𝔽13) generated by

300
010
001
,
300
030
003
,
100
040
0010
,
100
001
0120
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[3,0,0,0,3,0,0,0,3],[1,0,0,0,4,0,0,0,10],[1,0,0,0,0,12,0,1,0] >;

C32×Dic3 in GAP, Magma, Sage, TeX

C_3^2\times {\rm Dic}_3
% in TeX

G:=Group("C3^2xDic3");
// GroupNames label

G:=SmallGroup(108,32);
// by ID

G=gap.SmallGroup(108,32);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-3,90,1804]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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